Solubility of additive quartic forms over ramified quadratic extensions of Q₂

In this paper, we determine the minimal number of variables Γ - (d,K) which guarantees a nontrivial solution for every additive form of degree d = 4 over the four ramified quadratic extensions ℚ2(2), ℚ2(10), ℚ2(-2), ℚ2(-10) of ℚ2. In all four fields, we prove that Γ - (4,K) = 11. This is the first example of such a computation for a proper ramified extension of ℚp where the degree is a power of p greater than p.